This subproject is one of many research subprojects utilizing the resources provided by a Center grant funded by NIH/NCRR. Primary support for the subproject and the subproject's principal investigator may have been provided by other sources, including other NIH sources. The Total Cost listed for the subproject likely represents the estimated amount of Center infrastructure utilized by the subproject, not direct funding provided by the NCRR grant to the subproject or subproject staff. Distance distributions provide valuable structural information that could help to provide insight into the static structure of membrane proteins and conformationally heterogeneous water soluble proteins as well as fluctuations in these structures. Another important aspect of determining distance distributions is the case of more than two spins. For example, in fully-labeled KcsA channels there are four spin-labels, with two possible distances due to symmetry considerations. The two distances and the stoichiometry could be used as additional parameters that could help to determine distance distributions with a high degree of confidence. These distributions could then be used to find the change in distances upon channel gating when only a fraction of the channels is in the open state. T4-Lysozyme served as testing ground for determining methods useful for solving the inverse problem of finding the distributions from the DQC-ESR spectra. High-quality data were obtained on T4-L 65/135 and 61/135 mutants at 17GHz with use of deuterated solvents. The DQC signals with excellent SNR were recorded on a time-scale of 6s and analyzed by the Tikhonov regularization method. The shapes of the distributions in this work are consistent with the distributions found in our previous study of T4-L by trial and error. In the theoretical part of this work, regularization methods were shown to outperform SVD methods. A direct conversion of DQC-ESR signals into distance distributions by the Tikhonov regularization method was facilitated by obtaining the regularization parameter from the L-curve criterion.